Christian Pommerenke

Boundary Behaviour of Conformal Maps (eBook, PDF)

• Format: PDF

Produktbeschreibung

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain. In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The book is meant for two groups of users. (1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understand ing. They tend to be fairly simple and only a few contain new material. Pre requisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a). (2) Non-experts who want to get an idea of a particular aspect of confor mal mapping in order to find something useful for their work. Most chapters therefore begin with an overview that states some key results avoiding tech nicalities. The book is not meant as an exhaustive survey of conformal mapping. Several important aspects had to be omitted, e.g. numerical methods (see e.g.

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 300
• Erscheinungstermin: 9. April 2013
• Englisch
• ISBN-13: 9783662027707
• Artikelnr.: 53197937

Autorenporträt

There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. This subject which, while embedded in parts of mathematics considered pure, also has very important applications in electrical engineering, fluid dynamics (e.g air current flows around wing profiles etc.). In classical applications of conformal maps, the boundary tended to be smooth: this is not the case in many modern applications (e.g. for Julia sets). Pommerenke's new book gives a comprehensive and up-to-date treatment of this subject in a highly self-contained form, and including many exercises.

Inhaltsangabe

1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve Families and Capacity.- 10. Hausdorff Measure.- 11. Local Boundary Behaviour.- References.- Author Index.